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Theorem itg2val 19488
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2val  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Distinct variable group:    x, g, F
Allowed substitution hints:    L( x, g)

Proof of Theorem itg2val
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 xrltso 10667 . . 3  |-  <  Or  RR*
21supex 7402 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 9015 . 2  |-  RR  e.  _V
4 ovex 6046 . 2  |-  ( 0 [,]  +oo )  e.  _V
5 breq2 4158 . . . . . . 7  |-  ( f  =  F  ->  (
g  o R  <_ 
f  <->  g  o R  <_  F ) )
65anbi1d 686 . . . . . 6  |-  ( f  =  F  ->  (
( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
76rexbidv 2671 . . . . 5  |-  ( f  =  F  ->  ( E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
87abbidv 2502 . . . 4  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) } )
9 itg2val.1 . . . 4  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
108, 9syl6eqr 2438 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7387 . 2  |-  ( f  =  F  ->  sup ( { x  |  E. g  e.  dom  S.1 (
g  o R  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  )  =  sup ( L ,  RR* ,  <  ) )
12 df-itg2 19382 . 2  |-  S.2  =  ( f  e.  ( ( 0 [,]  +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
132, 3, 4, 11, 12fvmptmap 6987 1  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   {cab 2374   E.wrex 2651   class class class wbr 4154   dom cdm 4819   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Rcofr 6244   supcsup 7381   RRcr 8923   0cc0 8924    +oocpnf 9051   RR*cxr 9053    < clt 9054    <_ cle 9055   [,]cicc 10852   S.1citg1 19375   S.2citg2 19376
This theorem is referenced by:  itg2cl  19492  itg2ub  19493  itg2leub  19494  itg2addnclem  25958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-pre-lttri 8998  ax-pre-lttrn 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-itg2 19382
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